COMPUTING SCIENCE
Life Cycles
Are there periodic booms and busts in the diversity of life on Earth? Hear a tale of fossils and Fourier transforms
Brian Hayes
Twin Peaks
For my own first experiments with the analysis of the diversity
curve, I decided to retain the entire set of 36,000 genera rather
than discard the doubtful cases; this proved not to be a problem.
But my attempts to get along without the cubic detrending curve were
unsuccessful. Fitting the data to linear or exponential curves left
large residuals, producing a massive low-frequency lump in the
spectrum that swamped all other signals. I tried piecing together
two linear trends, with a hinge point where the slope changes in the
Cretaceous, but that didn't help much. I had to concede the point:
If you want to examine midrange frequencies in this data set, you
need to remove lower frequencies first. A cubic curve seems to be
the best way to do that.
When I finally got a result, it was not what I expected. I would not
have been surprised to see a spectrum identical to that of Muller
and Rohde; after all, I was working from the same data and following
similar procedures. I would not have been astonished to see
something totally nonsensical, stemming from a bug in my program.
But in fact my graph was very similar to theirs, with peaks in the
same positions, yet there was also a conspicuous difference: The
spikes at 62 and 140 million years had swapped amplitudes. The
140-million-year peak was the higher one, looming over its
shorter-period sibling.
Tracking down the source of this discrepancy took more than a week.
My suspicion focused first on the decision to include all the
genera, even those of doubtful provenance. But when I reran the
analysis with only well-resolved genera, the outcome was very
similar, with energy still concentrated in the 140-million-year peak.
Next I considered the main visual difference between the histograms
that I generated and those published by Muller and Rohde.
Randomizing the dates of origination and extinction yields a
smoother contour, without the stairstep profile created when all
changes come at substage boundaries. Maybe the sharp corners of the
stairsteps somehow shift energy into the higher-frequency band?
Again the facts proved me wrong. I applied a smoothing filter to rub
the corners off the Muller-Rohde curve, and another algorithm to add
sharper local transitions to my own histogram. The spectra were
unchanged, continuing to disagree about the relative heights of the peaks.
In the course of my struggles with this issue, I tried altering my
methods in a number of ways, and eventually wound up with a
diversity curve that appeared to match the Muller-Rohde curve in all
but a few local details—and yet still the two spectra
disagreed. Could such tiny disparities have large consequences? The
puzzle was solved by Rohde, who guessed the source of the trouble as
soon as I sent him a copy of my graphs. Sepkoski had cataloged a
handful of genera from the Vendian period, which preceded the
Cambrian. Because the Vendian record was sparse and fragmentary,
Muller and Rohde had excluded it from their analysis. I knew of this
decision, but I had neglected to snip away the long tail of Upper
Vendian stragglers from my version of the database. (They were
included in the Fourier analysis, but were invisible in the graphs I
had been drawing.) Rohde correctly deduced that the presence of
those extra data points, spread out over an interval of 23 million
years, would cause just the distortion I was seeing, reinforcing the
140-million-year wave and damping the 62-million-year one. Once I
truncated my histograms at the start of the Cambrian, the spectra
produced by my program matched the ones published by Muller and Rohde.
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